Contracting with Imperfect Commitment and the
Revelation Principle: The Single Agent Case
Econometrica - July, 2001
Helmut Bester and Roland Strausz
October, 2009
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October, 2009
1 / 11
Motivation
Revelation Principle.
Only applicable when the mechanism designer is able to credible
commit to any outcome of the mechanism.
Unrealistic requirement.
It fails without commitment: if agent reveals his type, the principal
can extract all his surplus. Agent anticiaptes this and he will not report
truthfully.
With limited commitment there is a gap in implementation theory
that arise because the derivation of optimal contracts cannot appeal
to the conventional revelation theory.
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October, 2009
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Related Litearture
Literature on contracting with limited commitment simply sidesteps
this problem by imposing arti…cial restrictions on the form of
contracts.
Freixas, Guesnerie, and Tirole (1985): principla constrained to linear
contracts and cannot use a revelation mechanism.
Dewatripont’s (1989): in the analysis of renegotiation-proof labor
contracts simply imposes IC restrictions wihrtout justifying them.
La¤ont and Tirole (1993): focus on the two-type case and restrict the
regulator to o¤ering a menu of two contracts
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October, 2009
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Model: Contracting without commitment (General Setting)
Contracting problem between a principal and a single agent in an
adverse selection environment
Solution determines an allocation z = (x, y ) 2 Z = X Y
X = all those decisions to which the principal can contractually
commit himself.
Y = all those decisions that are not contractible.
The principal has to select y 2 F (x ), when he is commited to the
decision x 2 X
Agent is privately informed about his type
t 2 T = ft1 , ..., ti , ..., tjT j g; 2 jT j < ∞
Agent’s payo¤ Ui (x, y ), continuous and bounded on Z .
Principal only knows the probability distribution of the ti :
γ = (γ1 , ..., γi , ..., γjT j ), with γi > 0, Σi γi = 1
Principal’s payo¤ Vi (x, y ), continuous and bounded on Z .
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October, 2009
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Model: Contracting without commitment (Mechanism)
Then, principal needs some kind of information from the agent.
Mechanism: Γ = (M, x )
Principal chooses a message set M,
He commits himself to x (m ) when the agent selects to send the
message m 2 M
Γ induce the following game:
Agent selects m 2 M, determining the contractually speci…ed decision
x (m ) 2 X .
The agent’s strategy is a mapping q : T ! Q, where Q is the set of
probabilities over the messages.
Principal observes m 2 M, he updates his beliefs pi (m ) about agent’s
type and then chooses y 2 F (x (m ))
The principal’s strategy is a function y : M ! F (x (m ))
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October, 2009
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Model: Contracting without commitment (Equilibrium
Concept)
Give Γ, the principal is constrained to the allocations that can be
obtained through the Perfect Bayesian Equilibria of this game
Expected payo¤s for the principal and the ti -type agent are:
Ui (q, y , x jM ) =
V (q, y , x jM ) =
Z
M
Ui (x (m ), y (m ))dqi (m ),
∑ γi
i
Z
M
Vi (x (m ), y (m ))dqi (m ),
for the agent
for the principal
To constitue a PBE, the functions (q, p, y ) have to satisfy three
conditions:
Principal’s strategy has to be optimal given his beliefs:
∑ pi (m)V i (x (m), y (m)) ∑ pi (m)V i (x (m), y 0 )
i
i
for all y 0 2 F (x (m )
...
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Model: Contracting without commitment (Equilibrium
Concept-Cont...)
...
Agent anticipates the principal’s behavior y and chooses q to maximize
his payo¤:
Z
M
Ui (x (m ), y (m ))dq i (m )
Z
M
Ui (x (m ), y (m ))dq i0 (m ) for all qi0 2 Q
Principal’s belief are consistent with Bayes’rule
(q, p, y , x jM ) is incentive feasible (IF) if (q, p, y ) is PBE given Γ,
and is incentive e¢ cient (IE) if it is IF and there is no IF
(q 0 , p 0 , y 0 , x 0 jM ) such that:
V (q 0 , y 0 , x 0 jM ) > V (q, y , x jM )
0
0
0
Ui (q , y , x jM ) = Ui (q, y , x jM )
(q 0 , p 0 , y 0 , x 0
and
for all ti 2 T
(q, p, y , x jM ) and
jM ) are payo¤ equivalent (PE) if
0
0
0
V (q , y , x jM ) = V (q, y , x jM ) and
Ui (q 0 , y 0 , x 0 jM ) = Ui (q, y , x jM ) for all ti 2 T
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October, 2009
7 / 11
Direct Mecnanism
Proposition 1: Let (q, p, y , x jM ) be IE. Then there exists an IF
(q 0 , p, y , x jM ) and a …nite set M 0 = fm1 , ..., mh , ..., mjM 0 j g with
jM 0 j jT j such that (q, p, y , x jM ) and (q 0 , p, y , x jM ) are PE.
Moreover, the vectors q 0 (mh ) = (q10 (mh ), ..., qi0 (mh ), ... ..., qj0T j (mh )),
h = 1, ..., jM 0 j , are linearly independent.
If a mechanism uses more messages than types, the the vectors
q (mh ) 6= 0 are necessarily linearly dependent. This means that some
messages are redundant.
Previous Proposition alows the principal to disregard message sets
that contain more messages than the agent’s type.
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October, 2009
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Direct Mecnanism (Cont...)
Proposition 2: If (q, p, y , x jM ) is IE, the there exists a direct
mechanism Γd = (T , x̂ ) and an IF (q̂, p̂, ŷ , x̂ jT ) such that (q̂, p̂, ŷ , x̂ jT )
and (q, p, y , x jM ) are PE. Moreover, q̂i (ti ) > 0 all ti 2 T .
We can no longer guarantee that the agent reveals his type with
certainty. Still truthful reporting is an potimal strategy for the agent
and he chooses this strategy with positive probability.
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October, 2009
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Optimal Contracts
Then, the principal’s contracting problem can be formulated as a
programming problem:
max
q,p,y ,x
∑ ∑ γi qi (tj )Vi (z (tj )),
i
subject to
j
Ui (z (ti ))
Ui (z (tj )),
(1)
Ui (z (ti ))
Ūi ,
(2)
[Ui (z (ti )) Ui (z (tj ))] qi (tj ) = 0,
y (ti ) 2 arg max ∑ pj (ti )Vj (x (ti ), y ),
(3)
pi (tj ) ∑ γk qk (tj ) = γi qi (tj ),
(5)
y 2F (x (ti )) j
(4)
k
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October, 2009
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Conclusions
This paper gives an extension of the revelation principle to situations
without commitment between a principal and a single agent.
This general result can be applied in di¤erent environments:
Unobservable actions: y may not be contractible because it is not
publicly observable.
Shrot-term contracts
Renegotiation: when principal receives the message m he may realize
that the contract x (m ) is ex post ine¢ cient.
Cheap Talk: when Z = Y
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October, 2009
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